X. Wang discovered the genus-2 curve X given by the equation

   y^2 = x^5 - x^4 + x^3 + x^2 - 2x + 1.

Conjecturally, the Jacobian J of X is isogenous to a 2-dimensional
quotient of J_0(188).  As far as I [William] know, this has not been
verified.  The field of generated by the endomorphisms of J defined
over Q is Q(sqrt(5)).  Let rhobar be the mod-5 representation attached
to J, so rhobar is supported by J[Sqrt(5)].   

Some observations one makes using, e.g., modular symbols algorithms:

1. The newform f in S_2(188) that is conjecturally attached to J has 
   q-expansion

      q + b*q^3 + (-2*b - 4)*q^5 + (-b - 5)*q^7 + ...

   where b^2 + 3*b + 1 = 0.

2. Fix the reduction map b |--> 1 in F_5.  
   The mod-5 a_p for the reduction of f are then
     [ <2, 0>, <3, 1>, <5, 4>, <7, 4>, <11, 3>, <13, 3>, 
       <17, 4>, <19, 4>, <23, 0>, <29, 0>, <31, 0>, <37, 2>, 
       <41, 4>, <43, 0>, <47, 1>, <53, 2>, <59, 3>, <61, 0>, 
       <67, 3>, <71, 0>, <73, 3>, <79, 3>, <83, 0>, <89, 1>, 
       <97, 2> ]

3. There is no eigenform in S_4(188;F_5) that lies in
          ker (T_{23}) intersect ker (T_{29}).

** Attempt to verify the conditions of Savitt's theorem. **

The field K is the quadratic ramified extension Q_5(sqrt(5)) of Q_5.
The field k = F_5.  The represenation rho : G_Q --> GL_2(K) is the
Galois representation supported by Tate_5(J) tensor Q regarded as a
2-dimensional K-vector space.  Using the Weil pairing, I bet one can
show that det(rho) is the cyclotomic character (this is standard for
elliptic curves).  

It appears likely, but we have not proved, that the reduction rhobar
is absolutely irreducible, because so many of the a_p mod 5 are not
zero.   ASSUME this for now. 

We know that rhobar is modular by BCDT, as det(rhobar) is cyclotomic
and rhobar is irreducible. 

Because k=F_5, rhobar is actually a representation to GL_2(F_5).  
Because rhobar is ordinary, in the sense that a_5 mod 5 is nonzero,
[Deligne SLNM 179] implies that rhobar is of the form [alpha, *; 0, beta] 
with beta unramified and alpha*beta = chi.  Moreover, *=0 if and only
if the centralizer of rhobar is not F_5.  By the theory of companion forms,
*=0 if and only if there is a weight-4 mod-5 eigenform g such that
rho_g = rhobar tensor chi^3.  However, as remarked above (Observation 3),
there is no such g.  Here we are using that rhobar is modular, so,
by the part of Serre's conjecture that has been proved, we KNOW that
rhobar arises from f, modulo the caveat that we haven't really computed
the Serre level of rhobar.  (But that should be standard these days.)

Because 5 does not divide 188, Savitt believes that rho|G_ell has good
reduction, hence is crystalline.  Hodge-Tate weight 0,1 comes from the
determinant character being cyclotomic.  So by a theorem of Breuil,
rho should be Barsotti-Tate.

OOPS!  Because it's Barsotti-Tate instead of only potentially
Barsotti-Tate, Savitt's theorem just doesn't apply.

At least we will have proved that rho is modular by a quasi-theorem of
Breuil and Mezard.

Other things to look at:
  1. Families: e.g., Peter Bending's sqrt(2).
  2. Reformulate "Potentially Barsotti-Tate" in terms of modular
     forms.   If ell exactly divides the level N of f, can rho_{f,ell} be
     potentially Barsotti-Tate.
  3. WD?

